Numerical Integration

Numerically integrating a function is a common engineering task. The SciPy module provides a number of functions for integration. A general-purpose tool used to solve integrals like this"

$$I=\int_{a}^{b}f(x)dx$$

is the quad() function. SciPy provides a series of functions for different kinds of integrations, for example, dblquad for double integration and tplquad for triple integration.

We use the following command to import the relevant functions from the integrate package:

from scipy.integrate import quad, dblquad, tplquad

The quad function takes a large number of optional arguments that can be seen using the help(quad) command.

The input arguments of the quad() function include the integrand f(x) and the limits of integration a and b. quad() returns a tuple with the first value being the numerical result and the second being the estimation of the numerical error in the result.

res, err = quad(f, a, b)

Single integration#

Let’s learn about the basic usage of the quad() function by solving a basic integration problem:

$$I=\int_{0}^{\pi/2}2sin(x)cos(x)\space dx$$

The answer we retrieved from the quad() function is quite close to the actual answer, which is $1$.

For simple functions, we can use a lambda function instead of explicitly defining a function for the integrand. Let’s look at an example of this below:

In the SciPy namespace, we use Inf instead of oo to refer to infinity.

Multiple integration#

Multiple integration works in the same way as single integration. Let’s look at some examples of double and triple integration.

For dblquad() and tplquad(), the integration limits of all inner integrals need to be defined as functions.

Double integration#

We will use the dblquad() function to compute the double integral.

$$I=\int_{0}^{2}\int_{0}^{3}x^2y\space dxdy$$

One important thing to notice is the integration limits. The order of arguments for f must correctly correspond to the order of the integration in dblquad(). This is given below:

Let’s look at an example of double integration below:

For the integration of y, we passed lambda functions as the limits, since these, in general, can be functions of x. As you can see, the value of the actual integral and the integral computed by dblquad() has a minimal margin of error, which is the order of $10^{-13}$.

Triple integration#

We will use the tplquad() function to compute the triple integral. Let’s use an interesting example: the volume of a sphere. The integral for the volume of sphere is:

$$V=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{R}r^2sin\theta\space dr d\theta d\phi$$

where $R$ is the radius of the sphere.

Important Note: The documentation of tplquad() states that the integration limits of the inner integrals should be provided as functions of the outer integration variables even if they happen to be constants. This makes sure that the limits are always written in general terms.

The order of the arguments in the definition of the function should correspond to the order of integrations.

Let’s compute the volume for a sphere with a radius 3:

• We passed lambda functions for the limits of the theta integration, since these, in general, can be functions of r.

• We passed lambda functions for the limits of the phi integration, since these, in general, can be functions of r and theta.

As you can see, the value of the actual volume and the volume computed by the tplquad() has a minimal error, which is the order of $10^{-12}$.

In the next lesson, we will learn about interpolation.